1. Field of the Invention
The invention refers to a controller unit for resetting a deflection of an oscillator excited with a harmonic oscillation, a device including such a controller unit, in particular a rotation rate sensor, as well as to a method for operating and for manufacturing such a controller unit.
2. Description of the Prior Art
Conventional control methods are tailored to control problems with constant or only slowly changing command variables. The value of a controlled process variable affected from a disturbance is kept close to a predetermined set point, or is updated as close as possible to a changing set point. Some applications (e.g. micromechanical rotation rate sensors for analysis of a Coriolis force) provide a control loop for resetting a deflection of an oscillator oscillating with its resonance frequency when stationary. A controller for such a control loop with a harmonic oscillation as command variable is conventionally designed such that a harmonic force signal exciting the oscillator is compensated so that the oscillator—apart form the harmonic oscillation corresponding to the command variable—performs as little movement as possible.
Typically, this feedback control problem is solved as illustrated in FIGS. 1A to 1D. FIG. 1A refers to a device 100 with a controlled system such as a mechanical oscillator 190, whose translational or rotational deflection is captured by a sensor 170. The oscillator 190 is supported or suspended such that it is movable along a direction of excitation and able to oscillate with a resonance angular frequency ω0. A harmonic force signal Fe acts on the oscillator 190 along the direction of excitation. A measurement signal from the sensor 170 reproduces the movement of the oscillator 190 along the direction of excitation. The movement of the oscillator 190 includes a resonance oscillation with the resonance angular frequency ω0, modulated by a force F (disturbance).
The measurement signal (system output signal) is fed to a controller unit 120 with a demodulator 122. In the demodulator 122, the system output signal is multiplied with a harmonic signal of frequency ω0 which is equal to the resonance angular frequency ω0 of the oscillator 190, wherein a baseband version of the system output signal as well as additional frequency conversion products are formed. A low pass filter 124 damps higher frequency components, in particular at the double resonance angular frequency 2·ω0 of the oscillator 190. The baseband signal is fed to the controller 126, which operates in the baseband, whose design and dimensions can be established by known controller design methods. The controller 126 is, for example, a continuous PI-controller. Due to its integral component, high stationary position can be achieved in case of a constant command variable.
The output of the controller 125 is multiplied (modulated) with a harmonic signal of frequency ω0 equal to the resonance angular frequency □0 of the oscillator 190 in a modulator 128. The modulation product is fed to an actuator 180 as a controller signal, the actuator executing according to the controller signal a force to the oscillator 190 that acts opposite to the deflection of the oscillator 190. With the resonance angular frequency ω0 and the damping s0 of the oscillator as well as with the amplification A and the system dead time TS of the system formed of the actuator 180, the oscillator 190 and the sensor 170, the transfer function of the oscillator 190 to be controlled is given by equation (1):
                              G          ⁡                      (            s            )                          =                                            A                                                                    (                                          s                      +                                              s                        o                                                              )                                    2                                +                                  w                  0                  2                                                      ·                          ⅇ                                                T                  S                                ·                s                                              =                                                    G                0                            ⁡                              (                s                )                                      ·                          ⅇ                                                -                                      T                    S                                                  ·                s                                                                        (        1        )            
In what follows it is assumed that the damping s0 of the oscillator 190 is much smaller than its resonance angular frequency (s0<<ω0), and that the oscillator 190 is excited altogether with the harmonic force signal Fe, which has a force amplitude superposing, respectively amplitude modulating an exciting oscillation with the resonance angular frequency ω0 of the oscillator:Fe=F·cos(ω0·T)  (2)According to FIG. 1B the actuator 180, the oscillator 190 and the sensor 170 of FIG. 1A can then be illustrated as a system with a summation point 191 and a transfer function G(s). A controller signal generated by the controller unit 120 is added to the harmonic force signal Fe at the summation point 191 and the transfer function G(s) acts on the sum signal according to equation (1).The low pass filter 124 which has to show sufficient damping at the double resonance angular frequency of the oscillator to damp the frequency conversion product sufficiently at 2·ω0, limits the bandwidth of the controller and hence its reaction rate with respect to changes in the force amplitude F.
FIG. 1C schematically illustrates the signal u(t) at the output of a continuous PI-controller with transfer function GR(s). A constant input signal xd(t) at the controller input generates a time proportional gradient u(t) at the controller output.
For a continuous PI-controller with amplification factor KP and the integral action coefficient KI the step response u(t) is produced by a step signal σ(t) as input signal according to equation (3):u(t)=(KP+KI·t)·σ(t).  (3)
By L-transformation of σ(t) and equation (3), the transfer function GR(s) follows:
                                          G            R                    ⁡                      (            s            )                          =                                            U              ⁡                              (                s                )                                                                    X                d                            ⁡                              (                s                )                                              =                                    s              ·                              (                                                                            K                      P                                        ·                                          1                      s                                                        +                                                            K                      I                                        ·                                          1                                              s                        2                                                                                            )                                      =                                          K                P                            ·                                                s                  +                                                            K                      I                                                              K                      P                                                                      s                                                                        (        4        )            
A pole at s=0 resulting from the integral component is characteristic for the continuous PI-controller. In a PI-controller used in connection with a controlled system of first order with a system function Gs(s), the system parameter Ks, and the boundary angular frequency ω1 is, according to equation (5),
                                          G            S                    ⁡                      (            s            )                          =                              K            S                    ·                      1                          s              +                              ω                1                                                                        (        5        )            then the controller parameter amplification factor KP and integral action coefficient KI are typically chosen so that the pole in the system function GS(s) (system pole) is compensated by the zero of the transfer function of the controller GR(s) (controller zero). Equating coefficients in the equations (4) and (5) results in a condition for the controller parameter given by the relation according to equation (6):
                              ω          1                =                              K            I                                K            P                                              (        6        )            
Equation (6) determines only the ratio of the amplification factor KP to the integral action coefficient KI. The product of the system transfer function GS(s) and controller transfer function GR(s) gives the transfer function of the corrected open loop Gk(s). As the system pole according to equation (5) and the controller zero according to equation (4) cancel, the transfer function of the corrected open loop Gk(s) is given by the relation according to equation (7).
                                          G            k                    ⁡                      (            s            )                          =                                                            G                S                            ⁡                              (                s                )                                      ·                                          G                R                            ⁡                              (                s                )                                              =                                    K              S                        ·                          K              P                        ·                          1              s                                                          (        7        )            
From the corrected open loop frequency response, the stability properties of the closed loop can be deduced via the Nyquist criterion. Because of the integral characteristics of the corrected open loop an absolute value characteristic results which declines with 20 db/decade. The phase always amounts to −90° for positive frequencies, to which application of the Nyquist criterion is typically limited. The phase characteristic is an odd function and has, at frequency 0, a 180° step from +90° for negative frequencies to −90° for positive frequencies. The transfer function Gw(s) for the closed loop generally results from that of the corrected open loop Gk(s) according to equation (8):
                                          G            w                    ⁡                      (            s            )                          =                                            G              k                        ⁡                          (              s              )                                            1            +                                          G                k                            ⁡                              (                s                )                                                                        (        8        )            
From equation (8) it follows that the transfer function Gw(s) for the closed loop is only stable when the locus of the corrected open loop neither encloses nor runs through the point −1 for 0≦ω<∞. One condition equivalent to this is that, at the transition of the absolute value characteristic of the corrected open loop through the 0 dB line, the phase of the corrected open loop is larger than −180°. As the phase is constant at −90° in the above case, the closed loop is thus always stable independent of the choice of amplification factor KP.
The bandwidth of the closed loop can be deduced from the frequency at the transition of the absolute value characteristic through the 0 dB line. The absolute value frequency response can be shifted via the amplification factor KP along the ordinate and, thus, the transition through the 0 dB line, respectively influencing the bandwidth that results from it.
FIG. 1D illustrates, for one example with a controlled system of first order with the boundary angular frequency ω1=2·π·100 Hz, a system parameter KS=ω1 and a PI-controller whose controller zero is chosen to compensate the system pole and whose amplification factor is KP=2. The figure shows the absolute value frequency responses of controlled system, controller, corrected open loop, and closed loop in the left column from top to bottom and, in the right column from top to bottom, the phase frequency responses of the controlled system, controller, corrected open loop, and closed loop. As can be seen from the diagram at the bottom left, the bandwidth of the open loop defined by the frequency at which the absolute value frequency response of the closed loop has dropped by 3 dB amounts to approximately 100 Hz.
The use of a classical PI-controller assumes a comparatively constant common variable. For this reason applications in which a harmonic common variable of almost constant frequency is to be controlled require a demodulator and a downstream low pass filter, which generate a corresponding baseband signal from the harmonic input signal.